Behavioral Analysis And Optimal Control Of Competitive Systems Of Biological Populations With Size-Structure

In order to maintain ecological balances, conserve bio-diversities and exploit renewablebiological resources, the evolution rules of biological species have to be investigated extensively.To this end, scholars have established a large number of mathematical models, most of which do notconsider individual’s size structure. Individual body size is defined as some characteristic indexes,such as volume, length, diameter, maturity or other physical or statistical features. For most speciesconcerned, such as forests, fishes, etc, body size has a decisive impact on individual’s survivality,fertility and determines their commercial values for human. In addition, some of size indicators (e.g.length, diameter, etc.) provide a kind of practical parameters for human in the development of thepopulation resources. For example, nets can be precisely designed in accordance with the minimaldiameter of fishes we shall harvest. Up to now, size-structured population models have receivedmore and more attention.This dissertation is concerned with a system of two competitive species with size structure,studies its dynamics (such as the existence of solutions, uniqueness, non-negativity, boundedness,stability, the continuous dependence of solution on control variables) and control problems (optimalharvesting). By means of (linear and nonlinear) functional analysis (e.g. fixed point principle,Mazur Theorem, tangent and normal cones), differential equations, and modern optimization theoryand other tools, some theoretical results are obtained, which provides the necessary scientificcriterion for the practical application of the model.The main work of this dissertation is as follows:In chapter II，a competitive model for two populations with size distribution is proposed andanalyzed. The first section sees a nonlinear model in the term of partial and integro-differentialequations. Section 2 proves a linear comparison principle, which paves the way for the subsequenttreatments. The third section deals with the well-posedness of the model, including existence anduniqueness of solution, non-negativity and boundedness, the continuous dependence of solutions onthe control variables.Chapter III discusses the equilibrium solutions of model and their stabilities. Section 1 givesthe kind of equilibrium solutions and the conditions on which equilibrium exists, section 2 analyzesthe asymptotic stability of the equilibria. In section 3 we take an example into consideration andsolve it numerically to describe the stability visually.Chapter IV is devoted to the optimal control strategy of the population model with sizestructure and competition. Section 1 mainly deals with optimal profits problem. Firstly, the basicmodel and the assumptions for model parameters are formulated. Then applying the Mazur theorem, we prove the existence of the optimal control. Finally, we derive the optimality conditions, viatechniques of tangent and normal cones. Section 2 consists of an optimal harvest problemconstrained by an ecological balance. After the statement of the model and the problem,Dubovitskii-Milyutin’s therom, a powerful result in optimization theory, is used to describe thecontroller.